Remark 3.2 In literature, T-E inequalities for the \(N\) -th step transition kernel and invariant measure are shown to be valid only under the assumption of uniform transportation constants, i.e., there exists some some \(L>0\)  such that \(\mathcal {P}(x,\cdot ) \in T_{1}(L) \) , for all \(x \in \mathcal {X}\)  (see [1]}).

